Trees and branches in Banach spaces

Authors:
E. Odell and Th. Schlumprecht

Journal:
Trans. Amer. Math. Soc. **354** (2002), 4085-4108

MSC (2000):
Primary 46B03

Published electronically:
May 20, 2002

MathSciNet review:
1926866

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Abstract: An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree of a certain type on a space is presumed to have a branch with some property. It is shown that then can be embedded into a space with an FDD so that all normalized sequences in which are almost a skipped blocking of have that property. As an application of our work we prove that if is a separable reflexive Banach space and for some and every weakly null tree on the sphere of has a branch -equivalent to the unit vector basis of , then for all , there exists a subspace of having finite codimension which embeds into the sum of finite dimensional spaces.

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Additional Information

**E. Odell**

Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712

Email:
odell@math.utexas.edu

**Th. Schlumprecht**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Email:
schlump@math.tamu.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-02-02984-7

Received by editor(s):
October 10, 2000

Received by editor(s) in revised form:
November 7, 2001

Published electronically:
May 20, 2002

Additional Notes:
Research supported by NSF

Article copyright:
© Copyright 2002
American Mathematical Society